\newproblem{lay:1_8_25}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 1.8.25}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
  Given $\mathbf{v}\neq\mathbf{0}$ and $\mathbf{p}$ in $\mathbb{R}^n$, the line through $\mathbf{p}$ in the direction of $\mathbf{v}$ has the parametric
	equation $\mathbf{x}=\mathbf{p}+t\mathbf{v}$. Show that a linear transformation $T:\mathbb{R}^n\rightarrow \mathbb{R}^n$ maps this line onto another line
	or onto a single point (\textit{a degenerate line}).
}{
  % Solution
	Let's define $\mathbf{y}=T(\mathbf{x})$ and check whether it is a line or not:
	\begin{center}
		$\begin{array}{rcll}
			\mathbf{y}&=&T(\mathbf{x})& \text{By definition of }\mathbf{x}\\
			          &=&T(\mathbf{p}+t\mathbf{v}) & \text{By linearity of }T\\
								&=&T(\mathbf{p})+tT(\mathbf{v})
		\end{array}$
	\end{center}
	If $T(\mathbf{v})\neq \mathbf{0}$, then $\mathbf{y}$ describes a line that goes through $T(\mathbf{p})$ in the direction of $T(\mathbf{v})$.
	If $T(\mathbf{v})=\mathbf{0}$, then $\mathbf{y}$ is a single point.
}
\useproblem{lay:1_8_25}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
